Method of characterizing interactions and screening for effectors

ABSTRACT

This invention enables high throughput detection of small molecule effectors of particle association, as well as quantification of association constants, stoichiometry, and conformation. “Particle” refers to any discrete particle, such as a protein, nucleic acid, carbohydrate, liposome, virus, synthesized polymer, nanoparticle, colloid, latex sphere, etc. Given a set of particle solutions having different concentrations, dynamic light scattering measurements are used to determine the average hydrodynamic radius, r avg , as a function of concentration. The series of r avg  as a function of concentration are fitted with stoichiometric association models containing the parameters of molar mass, modeled concentrations, and modeled hydrodynamic radii of the associated complexes. In addition to the r avg  value analysis, the experimental data may be fit/analyzed in alternate ways. This method may be applied to a single species that is self-associating or to multiple species that are hetero-associating. This method may also be used to characterize and quantify the association between a modulator and the associating species.

PRIORITY

This application claims priority to U.S. non-provisional ApplicationSer. No. 13/578,593 filed Mar. 11, 2013, which is the National StageEntry of International Application PCT/US11/26287 with an InternationalFiling Date of Feb. 25, 2011 which in turn claims priority to U.S.provisional application Ser. No. 61/310,133 filed Mar. 3, 2010, “Methodof high throughput detection of small molecule effectors of particleinteractions, as well as derivation of particle binding stoichiometryand equilibrium association constants.”

BACKGROUND

The high throughput detection of small molecule inhibitors and/orenhancers of particle interaction is desired in many fields of science.By “particle,” we refer to such objects as protein and polymer moleculestogether with their conjugates and co-polymers, viruses, bacteria,virus-like particles, liposomes, polystyrene latex emulsions,nanoparticles, and all such particles within the approximate size rangeof one to a few thousand nanometers. Dynamic light scattering providesan excellent analysis method for screening large chemical libraries,such as small molecule libraries of compounds, for effectors of particleinteractions. Such libraries are typically available at molecularscreening centers, such as the Scripps Research Institute MolecularScreening Center in Jupiter, Fla., the Broad Institute in Cambridge,Mass., the Molecular Screening Shared Resource centered at theUniversity of California, Los Angeles, and others. Small moleculelibraries of compounds may also be held by companies, privateindividuals, foundations, etc. Compound collections can exceed 300,000molecules that possess diverse architecture and function. Depending onthe particles used, the high throughput screening of chemical librariescan lead to a greater understanding of cellular function, the discoveryof new drugs, or any variety of nanotechnology-related innovations.

Additionally, libraries of macromolecules, such as a library of aproteins subjected to site directed mutagenesis at a large number ofsites, may also be screened to identify which residue(s) modulate/changeinteractions with the binding partner(s). Additionally libraries ofnanoparticles, such as gold particles or quantum dots, may also bescreened against binding partners using this method. Alternatively, anucleic acid fragment library could be screened against a protein todetermine which region the protein may bind to. The aforementionedscreen types could be done in the presence or absence of small moleculemodulators. Any particle type can potentially be screened in thismanner.

The detection and characterization of reversible associations ofparticles in solution is also essential in many areas of science. Forillustrative purposes, we shall focus specifically upon the interactionsof protein molecules and their conjugates, though the techniquesdisclosed will be equally applicable to all the other particle typeslisted. Whenever the word “molecule” is used within this specification,it will be understood that the word “particle” may be substitutedtherefore in most cases without any limitations implied upon theinventive methods described. The study and measurement of molecularassociations is important for many reasons; whether to gainunderstanding of cellular function or to develop and formulatepharmaceuticals or other biologically active materials. Essentially,most pharmaceuticals have functionality due solely to association withmolecules within the body, so the discovery and accuratecharacterization of these associations is a key element forpharmaceutical development.

Molecules of the same species may self-associate to form dimers,trimers, and higher order complexes, whereas molecules of differentspecies may associate with each other to yield complexes of variouscompositions. More than two particle types may combine to form acomplex. Such associations may be reversible or irreversible. Forreversible associations, the binding affinities are characterized by aunique equilibrium constant. The equilibrium constant specifies therelative concentrations of the complex(s) and the component monomers fora given set of conditions. According to Le Châtlier's principle, everyclosed system must eventually reach equilibrium. When reactants in areversible process are in excess of their equilibrium concentrations,the reaction proceeds to convert the reactants to products and achieveequilibrium. Alternately, when products are in excess, the reactionproceeds in a reverse direction to convert product to reactant and againachieve equilibrium. For the reaction of molecules A and B to form thecomplex AB, A+B

AB, the equilibrium association constant is written:

${K_{a} = \frac{\lbrack{AB}\rbrack}{\lbrack A\rbrack\lbrack B\rbrack}},$where the bracketed terms correspond to the corresponding concentrationsof the molecules A, B, and their complex AB. Although constant understable conditions, the equilibrium constant of a given association maychange in response to environmental factors, such as temperature, buffersalinity, or the presence of other factors modulating the interaction.

There are many techniques used to measure equilibrium constants andcharacterize molecular associations. However, the majority can detectonly tightly bound interactions, and require the tagging/labeling orimmobilization of one of the binding partners. As any modification ofthe molecule can potentially influence the interaction, techniquesimplementing free-solution testing are optimal. “Free-solution” meansthat molecules are neither tagged/labeled nor immobilized for analysis.As no molecule-specific immobilization/tagging protocol is required,free solution techniques are not limited to a single molecular type,such as proteins. Free solution techniques are applicable to mostmolecular types.

There are several well-established free-solution methods to determinestoichiometry and equilibrium constants, such as calorimetry andsedimentation equilibrium. Static light scattering is another option.The theory of using static light scattering measurements at differentsolution concentrations to determine self or hetero associationconstants has long been known, cf. Huglin, 1972 Light Scattering FromPolymer Solutions, Academic Press, London and New York, by W. Burchardand J. M. G. Cowie in Section 17, Selected Topics in BiopolymericSystems, as well as Hirs, 1973, Methods in Enzymology Volume XXVII,Enzyme Structure, Part D, Academic Press, London and New York, by Pittzet al. in section 10, Light Scattering and Differential Refractometry.Such measurements were demonstrated fairly recently by T. Yamaguchi etal. in Biochem. Biophys. Res. Commun., 2002, Vol. 290, 1382-1387 andimproved upon by Attri et al., in Anal. Biochem., 2005, Vol. 346,132-138, where they termed the technique “concentration gradient lightscattering”.

In static light scattering, the intensity of scattered light isproportional to the molar mass of the molecule; a dimer scatters twiceas much light as two monomers. For example, in the study ofself-association, the static light scattering concentration gradientmethod measures the intensity of scattered light over a series ofconcentrations of the molecule studied. The scattered light changes foreach concentration, in accordance with the change in the population ofthe associated species. The association constant quantifies how theassociated species change at different concentration ratios. Todetermine the association constant and stoichiometry of the interaction,the experimental data are fit against models that estimate theconcentrations of the individual components present at each solutionconcentration.

The three free-solution methods, static light scattering, calorimetry,and sedimentation equilibrium, require a relatively large amount ofsample when used in their standard configurations. Techniques requiringminimal sample quantities for measurement are often required, as therequired molecules may not be available in sufficient quantities, assynthesis or isolation and purification can present significantchallenge and expense. Finally, none of the three methods are suitablefor high-throughput measurement. Thus, the low productivitycharacteristics of all three methods impede practical study spanning alarge compositional range.

Recently, a fourth free-solution method has been explored:back-scattering interferometry with high-throughput capability and verylow sample requirements. Cf Bornhop et al. Science 317, pages 1732-1736(2007). Unfortunately, this method is limited to systems that bind in a1:1 ratio. Other stoichiometries, which commonly occur in nature, cannotbe distinguished or characterized.

The search for a free-solution, high-throughput method with low samplerequirements and the ability to detect multiple binding stoichiometriesremains. To date, no such method has been reported. Our inventivemethod, on the other hand, based on the use of dynamic light scatteringresolves, thereby, the previously discussed problems.

Dynamic light scattering is a well-established technique, typically usedto determine the diffusion coefficients of scattering particles insolution and, from them, an associated set of hydrodynamic radii. Thehydrodynamic radius is the radius of a hard sphere whose diffusioncoefficient is the same as that measured for the sample particle.Dynamic light scattering, also known as quasi-elastic light scattering,or QELS, uses the measured fluctuations in the light scattered from asample to determine these quantities. When in solution, sample particlesare buffeted by the solvent molecules. This leads to a random motion ofthe particles called Brownian motion. As light scatters from the movingparticles, this random motion imparts a randomness to the phase of thescattered light, such that when the scattered light from two or moreparticles is combined, a changing intensity of such scattered light dueto interference effects will occur. The dynamic light scatteringmeasurement of the time-dependent fluctuations in the scattered light isachieved by a fast photon counter. The fluctuations are directly relatedto the rate of diffusion of the particles through the solvent. Thefluctuations are then analyzed to yield diffusion coefficients and, fromthese, the hydrodynamic radii of the sample.

The time variations of the intensity fluctuations are quantified bymeans of so-called autocorrelation techniques. Depending upon theexperimental configuration of the dynamic light scatteringinstrumentation, the resulting autocorrelation function may be anintensity-intensity or field-field autocorrelation function, or acombination of these two. The intensity-intensity correlation functionis

$\begin{matrix}{{g^{(2)}(\tau)} = \frac{\left\langle {{I(t)}{I\left( {t + \tau} \right)}} \right\rangle}{\left\langle {I(t)} \right\rangle^{2}}} & (1)\end{matrix}$where I(t) is the intensity of the scattered light at time t, and thebrackets indicate averaging over all t. The correlation function dependson the delay τ, that is, how the intensity variation in time t+τcorrelates to the intensity variation in t. FIG. 1 shows a typicalcorrelation function for a sample of Immunoglobulin G protein insolution. In this figure open triangles are data, and the solid line isa fit of the data to a simple exponential function, described below.

As described in various light scattering texts, cf. B. Chu, Laser LightScattering: Basic Principles and Practice, (Academic Press, Boston,1991), for a single particle freely diffusing in solution, thecorrelation function of Eq. 1 becomesg ⁽²⁾(τ)=B+β exp(−2Γτ)  (2)where B is the baseline of the correlation function at infinite delay(τ→∞), β is the correlation function amplitude at zero delay (τ=0), andΓ is the decay rate.

An algorithm is used to fit the measured correlation function to Eq. (2)to retrieve Γ. From this point, the diffusion coefficient for theparticle, D, is calculated from Γ from the relation,

$D = {\frac{\Gamma}{q^{2}}.}$Here, q is the magnitude of the scattering vector, i.e.

${q = {\frac{4\pi\; n_{0}}{\lambda_{0}}{\sin\left( {\theta/2} \right)}}},$where n₀ is the solvent index of refraction, λ₀ is the vacuum wavelengthof the incident light, and θ is the scattering angle. Finally, thehydrodynamic radius r_(h) of an equivalent diffusing sphere is derivedfrom the Stokes-Einstein equation,

${r_{h} = \frac{k_{B}T}{6\pi\;\eta\; D}},$where k_(B) is the Boltzmann constant, T is the absolute temperature,and η is the solvent viscosity.

Since it is relatively insensitive to stray background light from thewalls of the containing structures, dynamic light scatteringmeasurements may be made from very small sample volumes, thus reducingsample quantity requirements and enabling the use of high throughputmeasurements. As such, dynamic light scattering may be used withmicrotiter plate based systems or very small volume cuvettes, each suchsample holding element containing only a few microliters of sample.Although such measurements require a higher concentration of samplerelative to those needed for static light scattering measurements, thesmaller sample volumes typically result in a significant overallreduction in total sample quantity required.

The application of the static light scattering concentration gradientprocedures to dynamic light scattering, DLS, would be a significantimprovement for determining particle association stoichiometry andaffinity, as this would permit far smaller sample quantities, as well ashigh throughput processing. For the static light scattering method, aseach different sample composition is examined, an associated excessRayleigh ratio is measured. Such Rayleigh ratios are related directly tothe molecular species producing them. A dynamic light scatteringmeasurement, on the other hand, yields a correlation function derivedfrom the scattered light fluctuations attributed to these same molecularspecies. Such correlation functions may be decomposed, following certainassumptions, to represent the distributions, in terms of diffusioncoefficients and their associated hydrodynamic radii, of the scatteringmolecules.

Whereas static light scattering data are relatively easy to model interms of postulated associated states, DLS responses to the presence ofsuch states are far more complex. For example, the molar mass of amolecular homodimer scatters four times the amount of light as one ofits two monomers, or twice as much light as scattered by the twoseparated monomers. On the other hand, the difference of the diffusioncoefficient of a dimer from that of one of its composite monomersdepends critically upon the structure of the associated dimer.Considering just the corresponding hydrodynamic radii as a measure ofthese differences, there may be a range of values, whereas for thestatic light scattering case there is a known discrete value.

The application of dynamic light scattering, for obtaining molecularassociation constants and stoichiometries by modeling thereof a seriesof relative concentration gradients has been attempted infrequently overthe past four decades, although DLS techniques are frequently used tostudy irreversible particle association, particularly particleaggregation. Examples of such prior work are described by Claes, et al.in Chapter 5, An on-line dynamic light scattering instrument formacromolecular characterization, of Laser Light Scattering inBiochemistry Eds. S. E. Harding, et. al., 1992, The Royal Society ofChemistry, Cambridge, UK, and Wilson Journal of Structural Biology 142,56-65 (2003).

Self association was studied by Mullen et al., J. Mol. Biol. 1996, 262,746-755, MacColl et al., Biochemistry 1998, 37, 417-423, an Lunelli,Physical Review Letters 1993, 70(4), 513-516. In the dynamic lightscattering industry, Protein Solutions developed a “Fraction Calculator”in an early version of the Dynamics software to determine the fractionof each species in a binary equilibrium, using the average r_(h) andpostulating or measuring the two end points. Malvern also has recentlypublished an application note that proposes how the percent monomer in amonomer/dimer system can be estimated using the hydrodymic radius of themixture.

In terms of heteroassociations, Vannini et al., J. Biol. Chem., 2004,Vol. 279, Issue 23, 24291-24296, used dynamic light scattering topredict the stoichiometry of a protein complex by isolating the complexand estimating the molecular mass of the complex from the hydrodynamicradius. This study involved a complex that could be isolated from thecomponent protein monomers, indicating the association was irreversibleor very tightly bound. DLS measurements have also been performed at aseries of two or more ratios of two components such as the work by Wanget al., Biopolymers, 1981, v.20, p 155-168, Murphy et al., BiophysicalJournal, 1988, 54, 45-56, Leliveld S. R. et al., Nucleic Acids Research,2003, Vol. 31, No. 16, 4805-4813, and Sharma et al., BiophysicalJournal: Biophysical Letters, 2008, L71-L73.

BRIEF DESCRIPTIONS OF THE DRAWINGS

FIG. 1 shows an autocorrelation function derived from a solution ofImmunoglobulin G protein.

FIG. 2 compares two models used to estimate the hydrodynamic radius ofassociating species.

FIG. 3 shows results obtained from protein heteroassociation experimentsindicating binding and stoichiometry for chymotrypsin and soybeantrypsin inhibitor. No binding is found in the presence of the inhibitorAEBSF.

FIG. 4 shows results obtained from a protein heteroassociationexperiment indicating binding and stoichiometry for chymotrypsin andbovine pancreatic trypsin inhibitor.

FIG. 5 shows the results of a heteroassociation negative controlexperiment. The data are consistent with no binding between chymotrypsinand lysozyme.

FIG. 6 shows the results of a series of chymotrypsin self associationexperiments at differing solution salinities.

BRIEF DESCRIPTION OF INVENTION

As previously discussed, methods of observing particle association withDLS has been explored for some time, however, the further step ofmeasuring a comprehensive series of concentrations or ratios, followedby parametric modeling and calculation of association constants,stoichiometry, and conformations had neither been proposed nordemonstrated prior to the inventive methods described in thepublication, authored in part by the inventors, Hanlon et al.“Free-solution, label-free protein protein interactions characterized bydynamic light scattering” Biophysical Journal, 2010, v.98, p 297-304,and described herein. We also specify that the method described in thepaper may be used to identify the optimal ratio of two proteins forachieving the maximum amount of association, and thus the largestaverage solution hydrodynamic radius. This ratio can then be used toscreen for small molecule chemical inhibitors, as inhibition would thenresult in the largest possible change in solution hydrodynamic radius.Although the optimized ratio may often be the best choice for a largescale screen, other ratios may be used. This technique is also notlimited solely to screening of inhibitors, as other effectors/modulatorsmay also be identified with the inventive method described herein.

An additional benefit of this method is that it may be used topre-screen the chemical libraries for aggregation. Aggregated compoundscan act as “promiscuous” inhibitors, and yield false positive. Dynamiclight scattering is an excellent way of testing for these aggregates, asdocumented in Feng et al., Nature Chemical Biology, 2005, v.1, p.146-148.

High throughput screening for effectors which modulate the solutionviscosity by effecting changes on the interacting particles may also beidentified by this method, although an internal standard such anano-sized polystyrene sphere or other internal standard may be employedin the sample solution. DLS measurement of solution viscosity wasrecently detailed by He et al., “High throughput dynamic lightscattering for measuring viscosity of concentrated protein solutions,”Analytical Biochemistry, 2010, accepted for publication.

The key objective of the present invention is to provide a highthroughput method for identification of small moleculeeffectors/modulators of particle interactions through the screening oflarge chemical libraries, using dynamic light scattering to monitorchanges in hydrodynamic radius of optimized or non-optimized solutions.An additional objective is to provide a high throughput method ofscreening for said effectors using another aspect of DLS data, such as achange in the measured viscosity, time dependent changes, or some othervariable determined by DLS.

An additional objective is to provide a high throughput method ofscreening libraries of particles.

An additional objective of the inventive method disclosed is the meansby which all of the stoichiometry and association constants that areextracted from the static light scattering concentration gradientmethod, are derived equivalently from DLS measurements, with theadditional yield of associated species conformational information.

An additional objective of our inventive method is to provide means toreduce the total quantity of sample required for extraction of saidstoichiometry and association constants.

A further objective of our inventive method is the establishment of asuitable means by which the hydrodynamic radius of each complex may bemodeled using the hydrodynamic radii of its constituent molecules.

A further objective of our invention is to permit all measurements to bemade at considerably greater speed by means of a high throughput device.

Another objective of our inventive method is the ability to measure theassociation constants under a variety of different environmentalconditions such as temperature, storage periods, packaging, etc.

A further objective of our invention is to characterize the effects, ofsmall molecules that may modulate protein associations. Our inventionallows the characterization of the association constants between theassociating particles and their modulators.

An inventive free-solution method is described that uses dynamic lightscattering for the high throughput detection of small molecule effectorsof particle interactions. The method may also be used to screenlibraries of particles. The method may also be used to characterize theequilibrium association constants and the stoichiometry of reversiblecomplexes. The method is high-throughput with low sample requirements

For self-association, a series of solutions are made containingdifferent concentrations of the sample molecule, all in the samesolvent. For heteroassociation, the method begins with the mixing of twostock solutions, one of each molecule in the same solvent, at varyingratios beginning, for example, with 100% of the first molecule and 0% ofthe second molecule and ending with 0% of the first molecule and 100% ofthe second molecule. Any other series of varying ratios may be used, aswell. Additionally, for either self- or hetero-associationdeterminations, a modulator may be added to the solutions, and itseffects quantified.

Each member solution of the series so prepared is generally illuminatedwith a fine laser beam and the fluctuating light scattered by saidmember is processed to yield its corresponding autocorrelation function.The resultant autocorrelation function data, one data set for eachmember ratio prepared, are then processed according to the followingfour-step procedure: 1) Modeling the relative concentration of eachmolecular component in solution: The concentrations of all components ina solution at a given time are calculated from postulatedstoichiometries, association constant(s), K_(a), and the known a prioriconcentrations of the stock solutions of the constituent molecule(s). 2)Modeling the translational diffusion coefficient for associatedmolecules: Models describing the conformation of associating species areparameterized and then used to calculate the translational diffusioncoefficient, and thence its hydrodynamic radius, r_(h), for each of theself- or hetero-associated species. 3) Modeling DLS data based uponabove concentration and associated r_(h) models: DLS results arecalculated based on the known molar masses, the parameterized relativespecies concentrations, and the parameterized translational diffusioncoefficients. 4) Fitting modeled DLS results to DLS measurements: Theparameterized DLS results are then compared to the actual DLSmeasurements and a best fit of the parameters derived. A variety ofdifferent parameterized models of association constants may then becompared to a single datum or set of data. In this manner, the mostprobable parameter values are derived. Alternatively, of course, aspecific parameter set may be assumed ab initio, and the correspondingstoichiometry and association constant or constants of the complexes insolution determined.

Following the initial characterization, the particle mixturedemonstrating the highest hydrodynamic radius may be used for a largescale screen of chemical libraries, where a drop in the solutionhydrodynamic radius is a positive hit of the screen.

Alternately, the solution may be optimized to demonstrate an increase ofthe solution hydrodynamic radius to screen for an association promoter,or any change in signal as determined by the user to be the mostindicative of the desired effect.

In addition, libraries of modified/altered particles may be screenedagainst the reacting partner(s). The most substantial change in thesolution hydrodynamic radius could be interpreted as the most promisinghit.

In addition, multiple systems could be screened simultaneously.

DETAILED DESCRIPTION OF THE INVENTION

For illustrative purposes, we shall focus specifically upon theinteractions of protein molecules, though the techniques disclosed areapplicable to all the other particle types as specified in theBackground section of this specification. Again, whenever the term“molecule” is used, it will be understood that the word “particle” maybe substituted therefore in most cases without any limitations impliedupon the inventive method.

The method begins with sample preparation to be described presently. TheDLS data are collected, and then analyzed in a four-step procedure.First, the concentrations of all components in a solution at a giventime are calculated from postulated stoichiometries, associationconstant(s), K_(a), and the known a priori concentrations of the stocksolutions of the constituent molecule(s). Second, the translationaldiffusion constant, and hence its corresponding hydrodynamic radius, ofeach associated species is modeled. Third, the modeled concentrationdata and modeled hydrodynamic radius data are combined to model theexpected dynamic light scattering data at each sample ratio. Fourth, abest fit of the models to the collected DLS data is obtained. Thus, someor all parameters are adjusted to produce a best fit to the DLS data.Such fitting might be achieved using a least squares method, forexample. These four steps of data analysis are discussed in detail inthe following, and variations are possible, as would be apparent to oneskilled in the art of DLS or particle associations.

There are many different ways of preparing samples in varyingconcentrations or ratios, and there are many different DLS systemscapable of making the measurements required for our invention. A seriesof different concentrations, or ratios of two components, may be mademanually, or automatically. Automatic methods include commerciallyavailable fluid handling robots, inline dilution/concentration systems,automated multiple syringe systems, autosamplers with pre-treatmentcapability, etc. Below we outline one such sample preparation and onesuch measurement system based on the use of a high throughput methodusing microtiter plates.

In this example, to prepare the sample series for both self- andhetero-association, two stock solutions are mixed manually. For analysisof hetero-association, a high concentration, on the order of 0.5 mg/mL,solution of each pure molecule is prepared in the same solvent. Forlarger molecules, such as those greater than, say, 50 kDa, lowerconcentrations may be used. For analysis of self-association reactions,the molecular solution is prepared at the highest concentration to betested; the second solution is the pure solvent in which the samplemolecule was prepared. All solutions are filtered through a 0.02 μmfilter.

For either type of analysis, aliquots of the two solution mixtures aredispensed into a 1536 well microtiter plate in a series of ratios from100% A:0% B to 0% A:100% B, or some subset thereof, where the number ofratios prepared depends upon the desired detail of analysis. Typically,10-20 ratios are used.

Following the dispensing of the sample, the microtiter plate iscentrifuged at a rate and duration sufficient to remove any bubblespresent in the samples; typically 1000 g for 15 seconds. Wells may thenbe covered to avoid evaporation, for example, by dispensingapproximately 10 μL of paraffin oil into each well. The plate is thenre-centrifuged prior to being placed into a dynamic light scatteringinstrument programmed, for example, to make 25 one-second dynamic lightscattering measurements per well.

Other sample holding systems may be used for with this inventive method.Multiwell plates are only one possibility.

Step 1: Modeling Concentrations of all Components in Solution

For the reaction A+B

AB the equilibrium association constant is given as

$K_{AB} = {\frac{\lbrack{AB}\rbrack}{\lbrack A\rbrack\lbrack B\rbrack}.}$For known total molar concentrations of two species [A_(tot)] and[B_(tot)], and known or modeled association constant, the molarconcentrations of free solution unassociated [A] and [B] and molarconcentrations of associated species, such as [AB], may be derived byfitting the data to the model selected. DLS measurements are made ofeach mixture over a range of [A_(tot)]:[B_(tot)] ratios to providesufficient data to extract the reaction equilibrium constant in terms ofthe model selected. This basic approach may be applied generally tospecies that are reversibly self- and/or hetero-associating with anyassumed stoichiometry, as shown, for example by Cantor and Schimmel,Chapter 15, “Ligand interactions at equilibrium” in BiophysicalChemistry, Part III: The Behavior of biological macromolecules, W.H.Freeman and Company, New York, N. Y., 1980. Three examples are givenbelow.

EXAMPLE 1 Hetero-Association: A+BAB

Species A and B associating to form species AB with equilibriumassociation constant K_(AB):[A _(tot) ]=[A]+[AB]Equation [B _(tot) ]=[B]+[AB][AB]=K_(AB[A][B])which reduces to:

$\mspace{20mu}{\lbrack A\rbrack = {{\frac{\left\lbrack A_{tot} \right\rbrack}{\left( {1 + {K_{AB}\lbrack B\rbrack}} \right)}\lbrack B\rbrack} = {\frac{1}{2}\left\{ {\left( {\left\lbrack B_{tot} \right\rbrack - \left\lbrack A_{tot} \right\rbrack} \right) - {\frac{1}{K_{AB}}\left( {1 - \left\{ {1 + {2{K_{AB}\left( {\left\lbrack B_{tot} \right\rbrack + \left\lbrack A_{tot} \right\rbrack} \right)}} + {K_{AB}^{2}\left( {\left\lbrack B_{tot} \right\rbrack - \left\lbrack A_{tot} \right\rbrack} \right)}^{2}} \right\}^{\frac{1}{2}}} \right)}} \right\}}}}$

EXAMPLE 2 Self-Association: A+AAA

Species A self-associating to form species AA with equilibriumassociation constant K_(AA):[A _(tot) ]=[A]+2[AA][AA]=K_(AA)[A]²Which reduces to:

$\lbrack A\rbrack = {\frac{1}{4K_{AA}}\left( {\left\{ {1 + {8{K_{AA}\left\lbrack A_{tot} \right\rbrack}}} \right\}^{\frac{1}{2}} - 1} \right)}$

EXAMPLE 3 Three Binding Components: A+BAB, A+CAC

Species A, B, and C, associating to form species AB and AC withassociation constants K_(AB) and K_(AC). This example pertains toassociation modulators. Consider species A and B to be the primaryassociating species, and species C to be a modulator of thoseassociations, e.g. a small molecule inhibitor. In this case, thepresence of [AC] reduces the availability of free [A] in solution, andso reduces the quantity of [AB] in solution. The molar concentrations ofall species may be found using the set of equations:[A _(tot) ]=[A]+[AB]+[AC][B _(tot) ]=[B]+[AB][C _(tot) ]=[C]+[AC][AB]=K_(AB)[A][B][AC]=K_(AC)[A][C]The above five equations may be reduced to a set of three equations andthree unknowns, and solved.

To one skilled in the art it is clear that using the above technique wemay model the concentrations in solution for any combination of A and Bwith any stoichiometry. It is also clear to one skilled in the art thatany number of species inter-associating may be similarly modeled, andthat self-association may be modeled simultaneously withheteroassociation.

Step 2: Modeling Translational Diffusion Coefficient for AssociatedSpecies

For species that are associating, the net size of the associatingspecies will be larger than the size of the individual componentspecies, and the corresponding translational diffusion coefficient willbe smaller than that of the component species. In general, it is notpossible to exactly calculate the size and shape of the associatingspecies, although in special cases additional information may provide anestimate of the size, or constrain the possible sizes. There are manyways to model the hydrodynamic radius of associating species forexample, those shown in FIG. 2 and several of which are discussed below:

1) Assume hard spheres. For combinations of hard spheres touching atsingle points, the translational diffusion coefficient and correspondinghydrodynamic radii may be calculated numerically as shown by J. G. de laTorre and V. A. Bloomfield in Biopolymers, Vol. 16, 1747 (1977). Forexample, for an association consisting of two hard spheres of equalradii r touching at a point, the translational diffusion coefficient ofthe associating object is found to be ¾ of the translational diffusioncoefficient of one of the constituent objects. The hydrodynamic radiusfor that object is therefore 1.33 times the hydrodynamic radius for asingle hard sphere. A model assuming hard sphere associations with alinear conformation will give the maximum reasonable hydrodynamic radiusfor a composite object, unless by associating with each other the basicshape of the constituent objects change.

2. Assume “droplet”—the mass changes to some power of the radius. Thevolume and radius of a sphere are related by the relation v=4/3πr³. Fortwo spheres of radii r₁ and r₂ which associate into one large spherewhere the volumes are conserved, the resulting radius of the largesphere will be r={r₁ ³+r₂ ³}^(1/3). This “droplet” model of association,where the constituent species act as droplets of fluid combining to forma larger droplet, results in the most compact possible associatingstructure and the smallest possible change in hydrodynamic radius. Giventhis model, for two constituent objects having the same radius, theradius of the composite object will be 2^(1/3)=1.26 times the radius ofthe constituent objects. While many systems may be well approximated bydroplets coming together, others clearly will not. However, the increasein the hydrodynamic radius may be modeled in a similar way for objectsthat do not have the density of hard spheres. Proteins which are folded,for example, are found to generally have a hydrodynamic radius whichvaries as the molar mass to the power of 1/(2.34), rather than the powerof ⅓ as would be expected for hard spheres as described by Claes, et al.in Chapter 5 of Laser Light Scattering in Biochemistry Eds. S. E.Harding, et. al., 1992, The Royal Society of Chemistry, Cambridge,UKHarding et al. Proteins are coiled and folded, rather than beingsolid, and so this result is not surprising. Proteins which areassociating with one another will likely not form as compact a structureas a protein which simply folds, and so it is reasonable to assume thatassociating proteins could have a hydrodynamic radius which varies asthe molar mass to the power (1/a), where a is a number less than 2.34.By this line of reasoning, we may model the hydrodynamic radius of anassociating species as

$r = \left\{ {\sum\limits_{i}r_{i}^{a}} \right\}^{\frac{1}{a}}$where a may be fixed to some value, or may be allowed to vary as aparameter when fitting. From the above discussion it is clear that theway in which constituent objects associate has a significant bearing onthe hydrodynamic radius of the composite object. For two objects ofequal radii r associating, the hard sphere model yields 1.33r, thedroplet model with a=3 yields a hydrodynamic radius for the compositeobject of 1.26r, and the droplet model with a=2 yields 1.41r. Wedemonstrate below that the inventive methods may be used to estimate theway in which species associate, and so provide valuable informationregarding the conformation association.

Step 3: Modeling DLS Data Based Upon Above Concentration and Associatedr_(h) Models

DLS measurements determine either the field-field or theintensity-intensity autocorrelation function, for a single or multiplespecies in solution, as described by Chu, in sections 3 and 4 of LaserLight Scattering, Basic Principles and Practice (2nd Ed., DoverPublications, Mineola New York (2007)). For a single species insolution, the functions are relatively simple. Polydisperse solutions,i.e., solutions containing non-identical species, have autocorrelationfunctions with greater complexity.

The field-field autocorrelation function, g⁽¹⁾(τ), of a single speciesundergoing thermal translational diffusion (Brownian motion) is a simpleexponential function given asg ⁽¹⁾(τ)=exp(−Γτ)where τ is the autocorrelation delay time, Γ is the decay rate given byΓ=q²D  (3)where D is the translational diffusion coefficient, and q is thescattered wave vector given by q=(4πn/λ₀)sin(θ/2), where n is thesolvent refractive index, λ₀ is the vacuum wavelength of the light usedin the measurement, and θ is the scattering angle. For a sphericalobject of radius r, the translational diffusion coefficient is given bythe Stokes-Einstein relation

$\begin{matrix}{D = \frac{k_{B}T}{6\pi\;\eta\; r}} & (4)\end{matrix}$where k_(B) is the Boltzmann constant, T is the absolute temperature,and η the solution viscosity. The above theory is detailed by F. Reif insection 15.6 of Fundamentals of Statistical and Thermal Physics(McGraw-Hill, New York (1965)). The hydrodynamic radius r_(h) measuredin a DLS experiment is the radius of a sphere having the sametranslational diffusion coefficient as the species under study, and assuch is considered an equivalent spherical radius.

The intensity-intensity autocorrelation function, g⁽²⁾(τ), for a singlespecies in solution is related to the field-field autocorrelationfunction by the Seigert relationg ⁽²⁾(τ)=1+β[g ⁽¹⁾(τ)]²where the amplitude β is related to the number of coherence areas viewedin a measurement volume. For a single species in solution, theintensity-intensity autocorrelation function therefore becomesg ⁽²⁾(τ)=1+β exp(−2q ² Dτ).By measuring the normalized photon count autocorrelation function, theintensity-intensity autocorrelation function is obtained. For a singlespecies solution, the data may be fit to a simple exponential function,as shown in FIG. 1, and the hydrodynamic radius may be extracted fromthe decay rate.

For polydisperse solutions, the analysis becomes more complex. In thiscase the field-field autocorrelation function is the sum over the decayrates of all the species in solution, weighted by the relative amount oflight scattered into the detector by each species, such that

$\begin{matrix}{{g^{(1)}(\tau)} = {\int_{0}^{\infty}{{G(\Gamma)}{\exp\left( {{- \Gamma}\;\tau} \right)}{\mathbb{d}\Gamma}}}} & (5)\end{matrix}$Here, G(Γ)dΓ is the fraction of scattered light intensity due to specieswith decay rates from Γ to Γ+dΓ. The intensity-intensity autocorrelationfunction therefore becomes:

$\begin{matrix}{{g^{(2)}\tau} = {1 + {\beta\left\lbrack {\int_{0}^{\infty}{{\mathbb{e}}^{- \Gamma_{\tau}}{G(\Gamma)}\ {\mathbb{d}\Gamma}}} \right\rbrack}^{2}}} & (6)\end{matrix}$The relative contributions to G(Γ) by the different species are given bythe relative intensities of light scattered by the different species.The intensity of scattered light for a species of a particular molarmass M and mass concentration c is given by B. Zimm in J. Chem. Phys.,vol. 16, no. 12, 1093-1099 (1948) asR=K*McP(r _(g),θ)[1−2A ₂ McP(r _(g),θ)]  (7)where R is the excess Rayleigh ratio, meaning the ratio of the lightscattered from the solute and the incident light intensity, correctedfor size of scattering volume and distance from scattering volume. P(θ)is the form factor or scattering function which relates the angularvariation in scattering intensity to the root mean square radius, r_(g),of the particle. A₂ is the second virial coefficient, a measure ofsolvent-solute and solute-solute interaction and is the second term inthe virial expansion of osmotic pressure. A₂ enters into the lightscattering equation as a correction factor for concentration effects dueto intermolecular interactions influencing the scattering lightintensity. M is the molar mass, c is the solute concentration in g/mL,and K* is defined as follows:

$K^{*} = {\frac{4\pi\; n_{0}^{2}}{N_{A}\lambda_{o}^{4}}\left( \frac{\mathbb{d}n}{\mathbb{d}c} \right)^{2}}$where n₀ is the solvent refractive index, N_(A) is Avagadro's number, λ₀is the vacuum wavelength of incident light, and dn/dc is the specificrefractive index increment.

For a distribution of species with a distribution of r_(h) andassociated distribution of Γ, with corresponding distributions of M(Γ),c(Γ), r_(g)(Γ), and A₂(Γ), using equations, 3, 4, 6, and 7, the expecteddistribution of exponential functions which would be observed in anintensity-intensity autocorrelation DLS measurement may be seen to be

$\begin{matrix}{{G(\Gamma)} = {\frac{R\left\lbrack {{M(\Gamma)},{c(\Gamma)},{r_{g}(\Gamma)},{A_{2}(\Gamma)}} \right\rbrack}{\int_{0}^{\infty}{{R\left( {{M(\Gamma)},{c(\Gamma)},{r_{g}(\Gamma)},{A_{2}(\Gamma)}} \right)}\ {\mathbb{d}\Gamma}}}.}} & (8)\end{matrix}$Equation (8) may be simplified for some cases. If all species involvedin the measurement have root mean square radii about a factor of 50 ormore smaller than the wavelength of light in the solution being used formeasurement (i.e. r_(g)<10 nm for 660 nm light in water), then thescattering function P(θ) approaches 1.0 regardless of the angle ofmeasurement and may be disregarded. A second simplification may be madefor the case where for all species involved in the measurement2A₂McP(θ)<<1, enabling this term including to be neglected. With boththese assumptions, the intensity of scattered light from a singlespecies is given byR=K*Mc  (9)and Equation (8) simplifies to

$\begin{matrix}{{G(\Gamma)} = {\frac{{M(\Gamma)}{c(\Gamma)}}{\int_{0}^{\infty}{{M(\Gamma)}{c(\Gamma)}\ {\mathbb{d}\Gamma}}}.}} & (10)\end{matrix}$

Inventive steps 1 and 2 described above generate a modeled distributionof species, each species having a particular concentration, molar mass,and hydrodynamic radius with associated decay rate. Given that modeleddistribution, Equation (8), or if appropriate Equation (10), may becombined with Equation (5) or (6) to generate the modeled field-field orintensity-intensity autocorrelation function, respectively. UsingEquation (10), the intensity-intensity autocorrelation function may beseen to be

${g^{(2)}(\tau)} = {1 + {{\beta\left\lbrack \frac{\int_{0}^{\infty}{{M(\Gamma)}{c(\Gamma)}{\exp\left( {- {\Gamma\tau}} \right)}\ {\mathbb{d}\Gamma}}}{\int_{0}^{\infty}{{M(\Gamma)}{c(\Gamma)}\ {\mathbb{d}\Gamma}}} \right\rbrack}^{2}.}}$

Step 4: Fitting Modeled DLS Results to DLS Measurements.

Procedure A: Directly Fitting all Autocorrelation Functions

Performing inventive steps 1, 2, and 3 permit the calculation of theexpected autocorrelation functions for all samples measured. Allautocorrelation functions may thus be fit, either individually or inconcert, with association constants, conformation parameters, andhydrodynamic radii as parameters in the fit. In this way the parameterswhich most accurately represent the data may be determined. Any of theparameters in the fit may be fixed to known values or may be varied as apart of the fitting procedure.

Procedure B: Fitting G(Γ)

Alternatively, estimates for the distribution G(Γ) at each concentrationmay found from the autocorrelation data, and compared to the modeledG(Γ) to determine the best fit parameters. It is not possible touniquely determine G(Γ) from data of g⁽¹⁾(τ) or g⁽²⁾(τ), and directcomparison between modeled and measured G(Γ) is not possible. However,it is possible to estimate G(Γ) from the autocorrelation function usingthe method of regularization, as discussed by S. W. Provencher inMakromol. Chem., vol. 180, 201-209 (1979), and developed further by manyothers.

Procedure C: Fitting Derived Quantities

The methods of fitting procedures A and B can be mathematically involvedand computationally intensive. Instead, each autocorrelation functionmay be analyzed separately, generating just one or two derived valueswhich contain most of the information characterizing the distributionG(Γ). Those derived quantities may then be compared to the valuesexpected by the modeling, and the modeled parameters may thus bedetermined. There are many functional forms used to fit individualautocorrelation functions. One class of functions generally used to fitautocorrelation function data is generated by assuming some functionalform for G(Γ), such as a Gaussian distribution, and calculating thecorresponding expected g⁽¹⁾(τ) or g⁽²⁾(τ). For the example of a Gaussiandistribution of G(Γ), the center and width of the Gaussian distributionare two of the free parameters used when fitting the autocorrelationfunction data. A second class of functions used to fit autocorrelationfunction data use an expansion of the distribution G(Γ) into moments ofthe distribution, and calculating the corresponding expected g⁽¹⁾(τ) org⁽²⁾(τ). The most commonly used implementation of this class offunctions is the method of cumulants expansion, as described by D. E.Koppel in J. Chem. Phys. 57, 4814-4820 (1972). The method of cumulantsmay be used to fit the autocorrelation data to determine the first andsecond cumulants, which are identical to the first and second moments ofthe distribution G(Γ). The first and second moments of this distributionare defined as

$\mu_{1} = {\overset{\_}{\Gamma} = {{\int_{0}^{\infty}{\Gamma\;{G(\Gamma)}\ {\mathbb{d}\Gamma}\mspace{14mu}{and}\mspace{14mu}\mu_{2}}} = {\int_{0}^{\infty}{\left( {\Gamma - \overset{\_}{\Gamma}} \right)^{2}\;{G(\Gamma)}\ {\mathbb{d}\Gamma}}}}}$respectively. The first moment is the mean decay rate, and is oftendesignated by the symbol Γ. The second moment is proportional to thewidth of the distribution G(Γ), and is often used in the definition ofthe polydispersity of a measured sample as

${Pd} = {\frac{\mu_{2}}{\mu_{1}^{2}}.}$Both the first and second moments of the modeled distribution G(Γ) maybe calculated for the models described above and fit to the values forthe first and second moment derived from data for all concentrations,and so the modeled parameters may be extracted. Below we provide anexample of this procedure for the case of comparing the mean decay ratebetween the data and models of association.

The mean decay rate Γ may be used to calculate an equivalent radiusspherical species, termed the average hydrodynamic radius, r_(avg).Given species A and B which satisfy the conditions making Equation (9)valid, having molar masses M_(A) and M_(B), concentrations c_(A) andc_(B), and hydrodynamic radii r_(hA) and r_(hB), with associatedtranslational diffusion coefficients D_(A) and D_(B) and correspondingdecay rates Γ_(A) and Γ_(B), the function G(Γ) is given by

${G(\Gamma)} = \frac{{{\delta\left( {\Gamma - \Gamma_{A}} \right)}M_{A}c_{A}} + {{\delta\left( {\Gamma - \Gamma_{B}} \right)}M_{B}c_{B}}}{{M_{A}c_{A}} + {M_{B}c_{B}}}$where δ(x) is the Dirac delta function having a value of 1 for x=0 and 0otherwise. The mean decay rate becomes

$\overset{\_}{\Gamma} = {{\int_{0}^{\infty}{\Gamma\;{G(\Gamma)}\ {\mathbb{d}\Gamma}}} = \frac{{\Gamma_{A}M_{A}c_{A}} + {\Gamma_{B}M_{B}c_{B}}}{{M_{A}c_{A}} + {M_{B}c_{B}}}}$Expressing the decay rates in terms of the equivalent hydrodynamic radiiand cancelling common terms, we find

$\frac{1}{r_{avg}} = \frac{{\left( {1/r_{A}} \right)M_{A}c_{A}} + {\left( {1 + r_{B}} \right)M_{B}c_{B}}}{{M_{A}c_{A}} + {M_{B}c_{B}}}$Where r_(avg) is the hydrodynamic radius corresponding to the mean decayrate Γ. This may be rewritten as

$r_{avg} = \frac{{M_{A}c_{A}} + {M_{B}c_{B}}}{\left( {M_{A}{c_{A}/r_{A}}} \right) + \left( {M_{B}{c_{B}/r_{B}}} \right)}$We may extended this analysis for an arbitrary number of species to

$\begin{matrix}{r_{avg} = \frac{\sum\limits_{i}\;{M_{i}c_{i}}}{\sum\limits_{i}\;{M_{i}{c_{i}/r_{i}}}}} & (11)\end{matrix}$Equation (11) is specific for the case of validity of Equation (9), butmay be expressed more generally by substituting Equation (7) forEquation (9) during the derivation.

Example of Steps 1-4:

As an example of the inventive method, we consider two species A and Bassociating to form species AB with equilibrium association constantK_(AB): A+B

AB. In this example we will use fitting procedure C, using the averagehydrodynamic radius found by fitting the autocorrelation functionmeasured for each sample to a cumulants model. We will assume thatEquation (11) is valid. The mass concentrations used in Equation (11)are given by multiplying the molar concentrations by the molar mass ofeach species, giving

$r_{avg} = \frac{{M_{A}^{2}\lbrack A\rbrack} + {M_{B}^{2}\lbrack B\rbrack} + {\left( {M_{A} + M_{B}} \right)^{2}\lbrack{AB}\rbrack}}{\left( {{M_{A}^{2}\lbrack A\rbrack}/r_{A}} \right) + \left( {{M_{B}^{2}\lbrack B\rbrack}/r_{B}} \right) + \left( {{\left( {M_{A} + M_{B}} \right)^{2}\lbrack{AB}\rbrack}/r_{AB}} \right)}$The modeled molar concentrations for A and B, and AB are given by thefollowing equations in terms of total molar concentrations [A_(tot)] and[B_(tot)] as

$\lbrack B\rbrack = {{\frac{1}{2}{\left\{ {\left( {\left\lbrack B_{tot} \right\rbrack - \left\lbrack A_{tot} \right\rbrack} \right) - {\frac{1}{K_{AB}}\left( {1 - \left\{ {1 + {2\;{K_{AB}\left( {\left\lbrack B_{tot} \right\rbrack + \left\lbrack A_{tot} \right\rbrack} \right)}} + {K_{AB}^{2}\left( {\left\lbrack B_{tot} \right\rbrack - \left\lbrack A_{tot} \right\rbrack} \right)}^{2}} \right\}^{\frac{1}{2}}} \right)}} \right\}\mspace{20mu}\lbrack A\rbrack}} = \frac{\left\lbrack A_{tot} \right\rbrack}{\left( {1 + {K_{AB}\lbrack B\rbrack}} \right)}}$and [AB]=K_(AB)[A][B]. Given hydrodynamic radii r_(A) and r_(B), forthis example we will assume r_(AB)={r_(A) ^(a)+r_(B) ^(a)}^(1/a). Ifmolar masses M_(A) and M_(B) and hydrodynamic radii r_(A) and r_(B) areassumed known, and [A_(tot)] and [B_(tot)] are known from samplepreparation, then association constant K_(AB) and the associationconformation parameter a are the only free parameters in a fit betweenthe data and the model.

The high throughput screening may follow the complete characterizationof the interaction. Alternatively, once the molar ratio/concentrationshowing the maximum hydrodynamic radius is determined, fullcharacterization may be bypassed if the sole point of interest isdiscovery of a protein protein interaction inhibitor/promoter/effector.Or, a ratio/concentration of the particles may be chosen based on othercriteria.

The library may be added before or after the addition of the proteinsolution to the plate. The optimal mixture of protein ‘A’ and protein‘B,’ may then be placed in the wells of a microtiter plate. Thehydrodynamic radius of each well would then be measured. The plate maybe measured in the DLS plate reader immediately, or after a set periodto allow the reaction with the library compounds to occur. Alternately,the plate could be scanned continuously over a set time period.Depending on the time constant of the potential inhibition, the changein the hydrodynamic radius may be possible to monitor through time, andthe kinetics of the interaction quantified. Sample holding systems arenot limited to multiwell plates. Other systems may be used as well.

Additional Application:

Measuring the equilibrium constant over a series of temperatures canyield thermodynamic information about the association: At equilibrium,there is no net change in the Gibbs free energy of a system, and therelationship of ΔG°, the free energy change of a reaction when all itsreactants and products are in their standard states, can be written as:ΔG°=−RT ln K_(eq)where R is the ideal gas constant, and T is absolute temperature.Substituting in the Gibbs free energy at constant temperature andpressure, where H and S reflect enthalpy and entropy, respectively:ΔG°=ΔH°−TΔS°, yields the manner in which the equilibrium constant varieswith temperature:

${\ln\; K_{eq}} = {{{- \frac{\Delta\;{H{^\circ}}}{R}}\left( \frac{1}{T} \right)} + \frac{\Delta\;{S{^\circ}}}{R}}$The derivation of this function as described by D. Voet et al.,Biochemistry, 2^(nd) Ed John Wiley & Sons, Inc., New York, N.Y., (1995)Chapter 3, affects the reasonable assumption that ΔH° and ΔS° areindependent of temperature. A plot of ln K_(eg) vs. 1/T yields astraight line of slope −ΔH°/R and an intercept of ΔS°/R. The plot, knownas a van't Hoff plot, enables the values of ΔH° and ΔS° to be determinedfrom measurements of K_(eq) at two or more temperatures.

Further Examples

Having fully described the invention above, the following examples aregiven solely for illustrative purposes and are not intended to limit thescope of the invention in any manner.

In FIG. 3 the soybean trypsin inhibitor (molar mass of 22 kDa) andchymotrypsin (molar mass of 25 kDa) heteroassociation experiment resultsare shown. The r_(h) is plotted as a function of the molar ratio of thetwo proteins. Circle symbols show the data, the solid line shows thefit. This data was fitted with an incompetent fraction of chymotrypsin,meaning that it was assumed that a portion of the chymotrypsin insolution is unable to associate with the soybean trypsin inhibitor. Thesoybean trypsin binding site on chymotrypsin is known to possess someheterogeneity, as shown by Erlanger et al. in their 1970 paper,“Operation Normality of α-Chymotrypsin solutions by a sensitivepotentiometric technique using a fluoride electrode”, AnalyticalBiochemistry, 33, 318-322. Incorporating an incompetent fraction intothe fitting allows for the exclusion of the non-participating enzymefraction. For this heteroassociation, the association constant was foundto be 3.8×10⁶M⁻¹, with a corresponding a value of 2.26. The bindingstoichiometry was found to be 2:1 chymotrypsin:soybean trypsininhibitor, in accordance with the known association ratio.

The additional data set in FIG. 3, represented by open squares, is thesame experiment repeated in the presence of 500 mM 4-(2-Aminoethyl)benzenesulfonyl fluoride hydrochloride (AEBSF). AEBSF, is a smallmolecule known to be a serine protease inhibitor which binds to thechymotrypsin active site, thus inhibiting the binding of the soybeantrypsin inhibitor. Since AEBSF is a small molecule (r_(h)<<1 nm), thedecay rate of the autocorrelation function associated with AEBSF isfaster than may be observed with conventional DLS technology, and thepresence of that molecule is not observed in the DLS signal. When AEBSFassociates with chymotrypsin, the r_(h) of the associated species is notmeasurable different from that of unassociated chymotrypsin. Theincrease in r_(h) seen in the absence of AEBSF is not observed in thisexperiment. This shows that the increase in r_(h) is due to specificsite binding of the two proteins, an interaction which is completelyinhibited by AEBSF. This negative control also demonstrates thepotential of the technique for large scale screening of small moleculemodulators of particle—particle associations. Additionally, theassociation constant of AEBSF with chymotrypsin may be measured usingthis invention by repeating the chymotrypsin/soybean trypsininhibitor/AEBSF experiment and lowering the concentration of AEBSF untilsome increase in r_(h) is seen, and modeling the associating species.

In FIG. 4 the bovine trypsin inhibitor (6.5 kDa) and chymotrypsinheteroassociation experiment results are shown. The r_(h) is plotted asa function of the molar ratio of the two proteins. Circle symbols showthe data, while the solid line shows the fit. For thisheteroassociation, the data were found to be consistent with anassociation constant of 6×10⁶ M⁻¹ with an a value of 3, and the bindingstoichiometry was found to be 1:1 chymotrypsin:bovine trypsin inhibitor,in accordance with the known association ratio. This associationconstant value closely matches the 6.3×10⁶ M⁻¹ value reported byKameyama et al., Biophysical Journal, Vol. 90, 2164-2169, (2006), whoanalyzed the two proteins in the same buffer conditions, using staticlight scattering. For comparison, a fit using association constants of0.1×10⁶ M⁻¹ is shown by broken lines. For these data, higher associationconstants (up to infinity) fit the data equally well as the solid lineshows, and so in this case it is possible only to provide a minimumvalue for the association constant.

In FIG. 5 the negative control of chymotrypsin and lysozyme (14.4 kDa)is shown. The r_(h) is plotted as a function of the molar ratio of thetwo proteins. Open square symbols show the data, while the solid lineshows the fit. Although the two proteins are oppositely charged underthe experimental conditions of pH 6.7, the association constant is foundto be 0, reflecting an absence of any specific interaction. This controlshows that only proteins with specific binding will be detected in thistechnique; non-associating proteins will not yield an associationconstant.

In FIG. 6, results of self-association experiments are shown. Here, theassociation constant for the dimerization of α-chymotrypsin isdetermined as a function of buffer salinity. On the left graph, ther_(h) is plotted as a function of the protein concentration. Opensymbols show the data, while solid lines indicate the fits. Squares,diamonds, triangles, circles, and stars represent data with solutionconcentrations of 50, 162.5, 275, 387, and 500 mM NaCl respectively.Note as the salinity increases, the r_(avg) increases, corresponding toan increase in association constant. The fitted association constantsextracted from each data set are graphed on the right, as a function ofsodium chloride concentration. Values closely match those determinedwith static light scattering, as reported by M. Larkin and P. Wyatt inchapter 8 of Formulation and Process Development Strategies forManufacturing of a Biopharmaceutical, John Wiley & Sons, Inc., New York,N.Y., in press 2008.

Many embodiments of this invention that will be obvious to those skilledin the art of dynamic light scattering measurements, particleinteractions, or high throughput screening are but simple variations ofthe basic invention herein disclosed. Accordingly,

We claim:
 1. A method to characterize and measure the equilibriumself-association constants of a molecular species and its underlyingstoichiometries within a given solvent comprising the steps of 1.collecting dynamic light scattering data from a plurality of samplesspanning range of varying concentrations of said molecular species insaid solvent comprising the steps of A. preparing a concentration seriesof a plurality samples of varying known concentrations of said molecularspecies in said given solvent; B. illuminating each member of saidconcentration series with a light beam from a laser source; C. measuringthe intensity fluctuations of light scattered by each said member ofsaid molecular concentration series; and D. deriving an experimentallyderived scattered light correlation function from each said measuredmember;
 2. modeling theoretical light scattering data comprising thesteps of A. postulating a plurality of possible stoichiometries of theself-associating species that may be present within each of said membersof said concentration series; B. parameterizing the concentrations ofeach self-associating and non-associating species within each member ofsaid concentration series; and C. parameterizing the hydrodynamic radiiof each self-associating species;
 3. comparing said experimentallyderived scattered light correlation functions from said plurality ofsamples to said modeled theoretical light scattering data for each ofsaid possible stoichiometries that may be present within each member ofsaid concentration series comprising the steps of A. calculating modeledtheoretical scattered light correlation functions from said modeledtheoretical light scattering data for each said postulated stoichiometryand said parametrized concentrations and said parametrized hydrodynamicradii; and B. obtaining a best fit of said experimentally derivedscattered light correlation functions from each said concentrationseries member to each of said modeled theoretical light scatteringcorrelation function, and
 4. deriving from said best fit the equilibriumassociation constants of said self-associations and their correspondingstoichiometries.
 2. The method of claim 1 where said concentrationseries is produced by the successive dilution of an initial highestconcentration of said molecular species, said dilution achieved using astock solution of said solvent.
 3. The method of claim 2 furthercomprising the step of adding a corresponding concentration of a secondmolecular species to some members of said concentration series.
 4. Themethod of claim 1 further comprising the step of modeling eachparameterized hydrodynamic radius of each complex by using thehydrodynamic radii of said complex's constituent molecules.
 5. Themethod of claim 1 wherein said concentration series consists of 10 to 20members.
 6. The method of claim 1 wherein said postulated hydrodynamicradii are fixed values determined a priori.
 7. The method of claim 1wherein said postulated concentrations are fixed values determined apriori.